Pollaczek polynomials and summability methods
Pollaczek polynomials satisfy a three-term recurrence formula for which the recurrence coefficients are rational functions of the degree n of the polynomial. They are important because they furnish an explicit example of orthogonal polynomials on [-1,1] for which the asymptotic theory of G. Szegö is not valid. Some Pollaczek polynomials are also important for the radial Coulomb problem in quantum mechanics. In both cases an asymptotic formula for the polynomials gives a lot of information. Such an asymptotic formula is usually obtained by using Darboux's method. The present paper gives an alternative approach based on regular transformations of sequences. The asymptotic formula holds whenever the variable is real and outside the oscillatory region [-1,1].
scientific article; zbMATH DE number 4152834